IMAGE COMPRESSION BY USING WALSH AND FRAMELET TRANSFORM

In this paper, Framelet and Walsh transform are proposed for transformation, and then using arithmetic coding for compress an image. The goal is to achieve higher compression ratio by applying two levels Framelet transform (FLT), and then apply 2D Walsh-Hadamard transform (WHT) on each 8x8 block of the low frequency sub-band, while all other sub-bands are ignored. Experimental results show that the proposed algorithm gets best possible solution for tradeoff between compression ratio (size of image) and quality of compressed image, Peak Signal to Noise Ratio (PSNR). The simulation was carried using MATLAB software package version 2014. In this work, experiments were carried out on the gray scale and colored images


INTRODUCTION
Image compression is an inevitable solution for image transmission since the channel bandwidth is limited and the demand is for faster transmission (Alsayyh and Dzulkifli, 2012).
The two major types of compression algorithms: lossless compression and lossy compression.
Lossless compression is used for applications that require an exact reconstruction of the original data, while lossy compression is used when the user can tolerate some differences between the original and reconstructed representations of the data. In return for accepting this distortion in the reconstruction, we can generally obtain much higher compression ratios than is possible with lossless compression (Khalid Sayood, 2006). Transform based image coding is one of the popular image compression method. Transform when applied on images; change the image pixels to frequency domain coefficients. Desirable property of transforms is that most of the image energy is concentrated only in few significant transform coefficients. Retaining these significant coefficients and eliminating remaining coefficients results in image compression.
Discrete Cosine Transform (DCT) and wavelet transform are commonly used transform methods for image compression. They are used in JPEG and JPEG 2000 respectively (Kekre et al., 2014). Recent advancements in this area show that transform based coding combined with other compression method results in better performance. In this paper, Framelet transform is used with Walsh transform for image compression. Framelet is very similar to wavelets, but has some important differences. Framelet has two or more high frequency filter banks, which produces more sub bands in decomposition. This can achieve better time-frequency localization ability in signal processing. Moreover, Framelet is more robust (Jiao and Lin, 2010). And the Walsh-Hadamard transform (WHT) is an invertible linear transform and is widely used in many practical image compression systems because of its compression performance and computational efficiency (Thida and Aye, 2015). The elements of the basis vectors of the Hadamard Transform take only the binary values ±1 and are, therefore, well suited for digital hardware implementations of image processing algorithms. Hadamard transform offers a significant advantage in terms of a shorter processing time as the processing involves simpler integer manipulation (compared to floating point processing with DCT) and the ease of hardware implementation than many common transform techniques. So it is computationally less expensive than many other orthogonal transforms (Veeraswamy and Srinivaskumar, 2017). In this work, will be applying 2D Walsh-Hadamard transform on each 8x8 block of the low frequency sub-band of Framelet transform, while all other sub-bands are ignored to get higher compression ratio. Such method is lossy compression method, then entropy coding of the quantized coefficients by Arithmetic coding (Arithmetic coding gives Kufa Journal of Engineering, Vol. 10, No. 2, 2019 29 greater compression than Huffman method, is faster for adaptive models, and clearly separates the model from the channel encoding (Moffat, et al, 1998). Quantization is the process of reducing the number of possible values of a quantity, thereby reducing the number of bits needed to represent it. Quantization is a lossy process and implies in a reduction of the color information associated with each pixel in the image (Thida and Aye, 2015).

DISCRETE WAVELET TRANSFORM (DWT)
In DWT, image is divided into four sub bands as shown in Fig. 1 a. These sub bands are formed by separable applications of horizontal and vertical filters. Coefficients that are represented as sub bands LH1, HL1 and HH1 are detail images while coefficients are represented as sub band LL1 is approximation image. The LL1 sub band is further decomposed to obtain the next level of wavelet coefficients as shown in Fig. 1b (Kumar and Agarwal, 2015).

FRAMELET TRANSFORM (FLT)
The three-channel filter bank, which is used to develop the FLT corresponding to a wavelet frame based on a single scaling function ɸ( ) and two distinct wavelets Ψ 1 (t) and Ψ 2 (t) the extra wavelet here makes this system an over complete one. It follows that ɸ( ) , Ψ 1 (t) and Ψ 2 (t) satisfies the dilation and wavelet equations (Choi, 2007).
The scaling function ɸ( ) and the wavelets Ψ 1 (t), Ψ 2 (t) are defined through these equations by the low-pass (scaling) filter h 0 ( ) and the two high-pass (wavelet) filters h 1 ( ) and h 2 ( ) , where the two distinct wavelets Ψ 1 (t) and Ψ 2 (t) are specifically designed to be offset from one another by one half as follows: Where the filters h 1 ( ) and h 2 ( ) should satisfy the Perfect Reconstruction (PR) condition.
This means that the input and output of the two filters are expected to be the same (Choi, 2007).

Filter bank structure for 2D framelet
To perform the FLT on 2D matrix, the transform first should be alternatively applied to the rows, then to the columns of the resulting matrices. This gives rise to nine 2-D sub-bands, one of which is the 2-D low pass scaling filter, and the other eight of which make up the eight 2-D wavelet filters, as shown in Fig. 3 (Abdulkareem, 2012).

computational of FLT for 2-D signal using separable method
A separable 2-D FLT can be obtained by alternating between rows and columns i.e., it processes each row in order and then processes each column of the result. Non-separable methods work in both matrix dimensions at the same time. A 2-D separable transform is equivalent to two 1-D transforms in series. It is implemented as 1-D row transform followed by a 1-D column transform on the data obtained from the row transform. To compute a single level discrete framelet transform for 2-D signal using separable method, the next steps should be followed  The final framelet transformed matrix is equal to:

Computation of IFLT for 2-D signal using separable and non-separable methods
To reconstruct the original signal from the discrete framelet transformed signal, Inverse Framelet Transform (IFLT) should be used. The inverse transformation matrix is the transpose of the transformation matrix as the transform is orthogonal, so to compute a single level 2-D inverse discrete framelet transform using separable method, the next steps should be followed To compute a single level inverse framelet transform for 2-D signal using non-separable method, the next steps should be followed:  Fig. 4. Since is a low-pass filter ℎ 0 ( ) while both 1 and 2 are high-pass filters (ℎ 1 ( ) and ℎ 2 ( )) , the 1 1 , 2 1 , 1 2 and 2 2 subbands each has a frequency-dom ain support comparable to that of the subband in a DWT. A similar scheme creates the 1 , 2 , 1 and 2 subbands with the same frequency-domain support as the corresponding ( ) subbands of the DWT, but with twice as many coefficients. Finally, note that there is only one subbands with the same frequency-domain as the subbands in a DWT (Abdulkareem, 2012). In this work, 2-Level 2-Dframelet decomposition is applied on the original image, then apply Walsh transformation on low frequency domain and ignore all high frequency domains. Then apply arithmetic code. The decompressed image is reconstructed by applying inverse Walsh transform and inverse framelet transform.

COMPRESS AND DECOMPRESS COLOR IMAGES
This algorithm is proposed for compressing the color images. First the RGB colors images are converted into Y C b C r form, then applying proposed algorithm on each layer independently, this means each layer from Y C b C r are compressed as a gray scale image. Fig. 6 shows that proposed algorithm is applied on each Y C b C r layer. For decompression color images, apply decompression on each layer then collect all layers in one matrix Y C b C r and convert Y C b C r format to RGB color image. Fig. 7 shown some gray scale images with proposed algorithm and the results in Tables 1, 2, 3 and 4 shown the performance of PSNR and CR compared with other algorithms  (Verma and Kadian, 2014) 1.75 26.87 81.42    From the results tabulated in Table 2, the time taken in proposed algorithm is lower than in other algorithms and gives higher PSNR  Tables 3, 4 and 5, respectively,

RESULTS AND DISCUSSION
show that the proposed algorithm using Walsh and framelet transform gives the best results for compression ratio with maintaining good quality for images.
Finally testing the proposed algorithm on Lena colour image. The comparison between original and decompressed image is shown in Fig. 8:

Fig. 8. Comparison between Original Colour Test Image and its Decompressed Image.
Good image quality and good compression ratio for the color images are illustrate in Table 6 after applying proposed algorithm.